The pulsed NMR techniques described herein, and in the above-identified related patents and applications, involve the use of a pulsed burst or pulse of energy designed to excite the nuclei of a particular nuclear species of a sample being measured. The protons, or the like, of the sample are first precessed in an essentially static magnetic field. The precession thus is modified by the pulse. After the application of the pulse, there occurs a free induction decay (FID) of the magnetization associated with the excited nuclei. That is, the transverse magnetization associated with the excited nuclei relaxes back to its equilibrium value of zero, and this relaxation produces a changing magnetic field which is measured in adjacent pickup coils. A representation of this relaxation is the FID waveform or curve.
An NMR system described herein, and in the above-identified related patents and applications, takes measurements of physical properties of polymer materials (e.g., rubber, plastics, etc.) and relates those properties back to, for example, flow rates (e.g., melt index), crystallinity, composition, density, and tacticity by performing the following analysis methods. The NMR system is first calibrated with known samples by determining the physical types, properties, and quantities of the target nuclei in each known sample. Unknown samples are then introduced into the NMR system, and the system determines the physical types, properties, and quantities of the target nuclei in each unknown sample based on the calibration information.
The analysis methods performed by the NMR system involve decomposing an FID curve associated with a known sample into a sum of separate time function equations. Useful time function equations include Gaussians, exponentials, Abragams (defined herein as Gaussian multiplied by the quantity sin(.omega.t) divided by .omega.t), modified exponential (defined herein as Ce.sup.-z where C is a constant, z=(kt).sup..alpha., and .alpha. lies between 0 and 1 or 1 and 2), modified Gaussian (defined herein as Gaussian multiplied by the cosine of the square root of t), and trigonometric.
The coefficients of the time function equations are derived from the FID by use of a Marquardt-Levenberg (M-L) iterative approximation that minimizes the Chi-Squared function. This iterative technique is well-known in the art. Other known-in-the-art iterative techniques can be used instead of M-L including a Gauss-Jordan technique, a Newton-Raphson technique, and/or a "steepest descent" technique.
From the time functions, a set of parameters is calculated. Some of these parameters are ratios of the y-axis intercepts, decay times (or time constants of decay) for each of the time functions, and the cross products and reciprocals thereof. The sample temperature may form the basis for another parameter.
Statistical modeling techniques are then used to select a subset of these parameters to form a regression equation or model. Regression coefficients are then computed for this parameter subset. These regression coefficients represent the regression model which relates the parameter subset to the types, properties, and quantities of the target nuclei in the known sample.
After the NMR system has been calibrated with one or more known samples, unknown samples can be introduced thereinto.
When an unknown sample is introduced into the calibrated NMR system, the FID associated with the unknown sample is decomposed into a sum of separate time function equations. The coefficients of the time function equations are derived from the FID by use of the iterative M-L technique. From the time functions, parameters are calculated. The parameters are then "regressed" via the regression model to yield the types, properties, and quantities of the target nuclei in the unknown sample. That is, the measured parameters from the FID of the unknown sample are used with the regression model, and the types, properties, and quantities of the target nuclei in the unknown sample are determined. This information is related to a property of the sample-under-test such as density, xylene solubles, or flow rates (e.g., melt index, flow rate ratio, and/or melt flow).
It should be understood that the regression model is a multi-dimensional regression equation or model, and it may be linear or non-linear. Because it is multi-dimensional, it may not be graphically represented. As an example of a regression technique, consider that the grade point average of each of the students at a college were related to that student's SAT score and high school standing, forming a three dimensional space. The line formed is a "regression line" which may be graphed. A new student's grade point average may be predicted by placing the new student's SAT score and high school standing on the "regression line" and "reading" the grade point average.
Melt index (MI) has been defined for polyethylene as the flow rate obtained under condition 190/2.16 (note 19, ASTM, or American Society for Testing and Materials, No. D1238-90b). In general, MI is a measure of viscosity. Flow rate ratio (FRR) has been defined for polyethylene as a dimensionless number derived by dividing the flow rate at condition 190/10 by the flow rate at condition 190/2.16 (paragraph 8.3, page 396, ASTM No. D1238-90b). In some cases, the logarithm of FRR is used instead of FRR. Melt flow (MF) has been defined for polypropylene as the flow rate at condition 230/2.16 (ASTM No. D1238-90b).
For some polymer materials, the NMR system may produce unacceptable results. For example, for some polyethylene "grades" (i.e., polyethylene products having specified density and MI values), the standard deviation between the MI estimation obtained by the NMR system and the "actual" MI value (e.g., the MI value obtained by a manual laboratory ASTM method) can become unacceptable if the calibration is carried out over the entire product "slate" (i.e., the totality of product grades manufactured by a particular producer).